I think "no sooner... than" implies a more immediate occurrence than "hardly... when", but other than that, there is no big difference and they're pretty interchangeable.
E.g. "No sooner had I started eating than the doorbell rang." makes me think they had sat down, and perhaps lifted their fork to take a bite, when the doorbell rang.
"I had hardly started eating when the doorbell rang." makes me think they had sat down and eaten a little, though not much, when the doorbell rang.